The value of sum_{n=1}^{10} int_{-2n-1}^{-2n} | vedclass

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(D) Let $I = \sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. Consider the first integral t

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(D) Let $I = \sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. Consider the first integral term: $J_n = \int_{-2n-1}^{-2n} \sin^{27} x \, dx$. Let $x = -t$, then $dx = -dt$. When $x = -2n-1$, $t = 2n+1$. When $x = -2n$, $t = 2n$. So, $J_n = \int_{2n+1}^{2n} \sin^{27}(-t) (-dt) = \int_{2n}^{2n+1} -\sin^{27} t \, dt = -\int_{2n}^{2n+1} \sin^{27} t \, dt$. Substituting this back into the sum, we get: $I = \sum_{n=1}^{10} (-\int_{2n}^{2n+1} \sin^{27} x \, dx) + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$. $I = -\sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx = 0$.

The value of $\sum_{n=1}^{10} \int_{-2n-1}^{-2n} \sin^{27} x \, dx + \sum_{n=1}^{10} \int_{2n}^{2n+1} \sin^{27} x \, dx$ is equal to

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